Proposition 5.1: redistribution converse outer bound

Let $A,C,B,R$ be nonempty finite types, let $\psi\in\mathbb C^{R\times(A\times(C\times B))}$ have norm $1$, and let $Q,E\in\mathbb R$. Assume $Q\ge0$ and that, for every $\varepsilon,\delta>0$, there is a natural $N$ such that for every $n\ge N$ with $n\ge1$ there are natural numbers $q,k,l$ and encoder and decoder channels with the following properties. The encoder takes $A^nC^n$ and $k$ qubits to $A^n$, $l$ qubits, and a $q$-qubit message; the decoder takes that message, $B^n$, and $k$ qubits to $C^nB^n$ and $l$ qubits. A channel here is a map $X\mapsto\sum_jK_jXK_j^\dagger$ for a finite Kraus family satisfying $\sum_jK_j^\dagger K_j=I$. Starting with $\psi^{\otimes n}$ and $\Phi_k=2^{-k/2}\sum_{z\in\{0,1\}^k}|z,z\rangle$, these channels, applied in that order with $R^n$ unchanged, produce a matrix within trace norm $\varepsilon$ of the projector onto $\psi^{\otimes n}\otimes\Phi_l$, with $A^n$ retained by the encoder and $C^nB^n$ at the decoder, and satisfy $q\le n(Q+\delta)$ and $k-l\le n(E+\delta)$. The trace norm is $\|M\|_1=\operatorname{Re}\operatorname{tr}\sqrt{M^\dagger M}$, and $q,k,l$ may be zero. Form $\rho_{uv}=\psi_u\overline{\psi_v}$; for a register collection $U$, reduce by summing over equal complementary labels and define $S(U)=-\operatorname{Re}\operatorname{tr}(f(\rho_U))$, where $f(t)=t\log_2t$ is applied by continuous functional calculus, with $0\log_20=0$. The conclusion is precisely